Finite Group Databasebeta

← #13155459

Subgroups — #13155459

Order 1728.

638 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88808C211yesmaximal, characteristic
2#11723437C211yesmaximal, characteristic
3C2×C3×#3712C211yesmaximal, characteristic
4#11722050C212yesmaximal, complemented
5C3×Dic3⋊(C3⋊C8)C212yesmaximal, complemented
6#10590695—32nomaximal
7#10590332—31nomaximal
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
10C2×C3×C3⋊(C3⋊C8)C2212yes—
11C32×C4×Dic3C2211yescharacteristic
12C2×C32×C3⋊C8C2212yes—
13#88808—22no—
14#88807C412yes—
15#88803C412yes—
16C3×#3712—22no—
17#64081—31no—
18C2×C3×ΓL(1,𝔽25)—32no—
19#63855—32no—
20#63855—61no—
21C2×#3712D311yescharacteristic
22C2×#3710—31no—
23C2×D3×C3⋊C8—32no—
24#63641—32no—
25C3×Dic3⋊C8—32no—
26C3×C4⋊(C3⋊C8)—32no—
27#63572—32no—
28#63572—32no—
29#63572—62no—
30Dic3⋊(C3⋊C8)D312yescomplemented
31C22×C3×C3⋊D3C2×C411yescharacteristic
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)C2×C412yes—
34C2×C3×C3⋊Dic3C2×C411yescharacteristic
35C3×C4×C3⋊D3D412yes—
36C3×C4×C3⋊D3Q812yes—
37C2×C32×Dic3C2×C412yes—
38C3×C3⋊(C3⋊C8)—22no—
39C32×C3⋊C8—22no—
40#5119—91no—
41#4839—92no—
42C2×C4×C3⋊D3C2×D311yescharacteristic
43C2×C3×C4×D3—32no—
44C2×C3×C4×D3—61no—
45C2×C32×C8—32no—
46C32×C42—31no—
47#3752—32no—
48#3750—32no—
49C4×C3⋊Dic3—31no—
50C4×C3⋊Dic3—32no—
51C4×C3⋊Dic3—62no—
52C4×C3⋊Dic3—61no—
53C4×C3⋊Dic3—121no—
54C2×C3⋊(C3⋊C8)C2×D312yes—
55C3×C4×Dic3—21no—
56C3×C4×Dic3C2×D312yes—
57C2×C3×C3⋊C8—22no—
58C2×C3×C3⋊C8—32no—
59C2×C3×C3⋊C8—32no—
60C2×C3×C3⋊C8—62no—
61C2×C3×C3⋊C8C2×D312yes—
62C2×C3×C3⋊C8C2×D312yes—
63C3×ΓL(1,𝔽25)—64no—
64Dic3⋊Dic3—32no—
65Dic3⋊Dic3—32no—
66Dic3⋊Dic3—62no—
67(C2×D3)⋊Dic3—32no—
68(C2×D3)⋊Dic3—32no—
69(C2×D3)⋊Dic3—62no—
70#3712—22no—
71#3710—62no—
72D3×C3⋊C8—64no—
73C22×C33C22×C411yescharacteristic
74C2×C3×C3⋊D3—22no—
75C2×C3×C3⋊D3C4⋊C412yes—
76C33×C4C2×D411yescharacteristic
77C33×C4C2×Q811yescharacteristic
78C3⋊(C3⋊Dic3)—22no—
79C3×C3⋊Dic3C4⋊C412yes—
80C32×Dic3—22no—
81C2×C3×M16—91no—
82#926—91no—
83C22×C3⋊C8—91no—
84C2×ΓL(1,𝔽25)—32no—
85C2×D3×C8—92no—
86#872—92no—
87#872—181no—
88C3×C4⋊C8—92no—
89Dic3⋊C8—32no—
90Dic3⋊C8—92no—
91Dic3⋊C8—92no—
92Dic3⋊C8—92no—
93Dic3⋊C8—182no—
94C4⋊(C3⋊C8)—32no—
95C22×C3⋊D3C4×D311yescharacteristic
96C22×C3×D3—32no—
97C22×C3×D3—61no—
98C2×C32×C4—22no—
99C2×C32×C4—21no—
100C2×C32×C4—32no—
101C2×C32×C4—41no—
102C2×C32×C4C22×D311yescharacteristic
103C2×C32×C4C22×D312yes—
104C2×C3⋊Dic3—32no—
105C2×C3⋊Dic3—32no—
106C2×C3⋊Dic3—32no—
107C2×C3⋊Dic3—62no—
108C2×C3⋊Dic3—62no—
109C2×C3⋊Dic3—62no—
110C2×C3⋊Dic3—122no—
111C2×C3⋊Dic3C4×D311yescharacteristic
112C4×C3⋊D3Dic612yes—
113C4×C3⋊D3#6712yes—
114C2×C3×Dic3—22no—
115C2×C3×Dic3—32no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3C4×D312yes—
118C2×C3×Dic3C4×D312yes—
119C3×C4×D3—34no—
120C3×C4×D3—34no—
121C3×C4×D3—62no—
122C3×C4×D3—62no—
123C32×C8—62no—
124C3⋊(C3⋊C8)—22no—
125C3×C3⋊C8—22no—
126C3×C3⋊C8—22no—
127C3×C3⋊C8—42no—
128C3×C3⋊C8—62no—
129C3×C3⋊C8—62no—
130C3×C3⋊C8—122no—
131#424—271noSylow
132C2×C33C2×C4⋊C411yescharacteristic
133C2×C33#13112yes—
134C3×C3⋊D3—22no—
135C22×C3×C4—91no—
136C2×C4×D3—32no—
137C2×C4×D3—61no—
138C3×M16—182no—
139C2×C3×C8—32no—
140C2×C3×C8—32no—
141C2×C3×C8—62no—
142C2×C3×C8—92no—
143C3×C42—32no—
144C3×C42—61no—
145#217—92no—
146(C2×D3)⋊C4—92no—
147(C2×D3)⋊C4—92no—
148(C2×D3)⋊C4—182no—
149C4⋊Dic3—92no—
150Dic3⋊C4—92no—
151Dic3⋊C4—92no—
152Dic3⋊C4—182no—
153C4×Dic3—92no—
154C4×Dic3—91no—
155C4×Dic3—182no—
156C4×Dic3—181no—
157C4×Dic3—361no—
158C4×Dic3D3211yescharacteristic
159C2×C3⋊C8—32no—
160C2×C3⋊C8—32no—
161C2×C3⋊C8—62no—
162C2×C3⋊C8—182no—
163C2×C3⋊C8D3212yes—
164ΓL(1,𝔽25)—64no—
165D3×C8—184no—
166C22×C32—22no—
167C22×C32—21no—
168C22×C32—41no—
169C22×C32C2×C4×D311yescharacteristic
170C22×C32C2×C4×D312yes—
171C2×C3⋊D3—22no—
172C2×C3⋊D3Dic3⋊C412yes—
173C2×C3×D3—34no—
174C2×C3×D3—62no—
175C2×C3×D3—62no—
176C2×C3×D3—62no—
177C2×C3×D3—122no—
178C32×C4—22no—
179C32×C4—21no—
180C32×C4—21no—
181C32×C4—22no—
182C32×C4—41no—
183C32×C4—41no—
184C32×C4—62no—
185C32×C4C2×Dic611yescharacteristic
186C32×C4D3×D412yes—
187C32×C4D3×Q812yes—
188C32×C4C2×#6711yescharacteristic
189C3⋊Dic3—62no—
190C3⋊Dic3—64no—
191C3⋊Dic3—62no—
192C3⋊Dic3—62no—
193C3⋊Dic3—122no—
194C3⋊Dic3—122no—
195C3⋊Dic3—124no—
196C3⋊Dic3Dic3⋊C412yes—
197C3×Dic3—22no—
198C3×Dic3—22no—
199C3×Dic3—34no—
200C3×Dic3—42no—
201C3×Dic3—62no—
202C2×M16—91no—
203C22×C8—271no—
204#117—271no—
205C4⋊C8—92no—
206C4⋊C8—272no—
207C33#42411yescharacteristic, complemented, Sylow
208C23×C3—91no—
209C22×D3—32no—
210C22×D3—61no—
211C2×C3×C4—22no—
212C2×C3×C4—21no—
213C2×C3×C4—32no—
214C2×C3×C4—32no—
215C2×C3×C4—41no—
216C2×C3×C4—62no—
217C2×C3×C4—91no—
218C2×C3×C4—92no—
219C2×C3×C4—92no—
220C2×C3×C4C2×D3212yes—
221C2×C3×C4C2×D3211yescharacteristic
222C2×Dic3—32no—
223C2×Dic3—61no—
224C2×Dic3—92no—
225C2×Dic3—92no—
226C2×Dic3—92no—
227C2×Dic3—182no—
228C2×Dic3—182no—
229C2×Dic3—182no—
230C2×Dic3—362no—
231C2×Dic3#62412yes—
232C4×D3—34no—
233C4×D3—34no—
234C4×D3—62no—
235C4×D3—62no—
236C3×C8—62no—
237C3×C8—62no—
238C3×C8—122no—
239C3×C8—182no—
240C3⋊C8—22no—
241C3⋊C8—62no—
242C3⋊C8—62no—
243C3⋊C8—122no—
244C2×C32—22no—
245C2×C32—22no—
246C2×C32—22no—
247C2×C32—22no—
248C2×C32—21no—
249C2×C32—41no—
250C2×C32—42no—
251C2×C32D3×C4⋊C412yes—
252C2×C32#90112yes—
253C2×C32#90712yes—
254C2×C32#90712yes—
255C2×C32C2×Dic3⋊C411yescharacteristic
256C3⋊D3—22no—
257C3×D3—64no—
258C3×D3—122no—
259C22×C4—91no—
260M16—182no—
261C2×C8—32no—
262C2×C8—92no—
263C2×C8—542no—
264C4⋊C4—272no—
265C22⋊C4—272no—
266C42—31no—
267C42—271no—
268C22×C3—21no—
269C22×C3—22no—
270C22×C3—41no—
271C22×C3—92no—
272C22×C3—182no—
273C22×C3C4×D3212yes—
274C22×C3C2×#62411yescharacteristic
275C2×D3—34no—
276C2×D3—62no—
277C2×D3—62no—
278C2×D3—62no—
279C2×D3—122no—
280C3×C4—22no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—22no—
284C3×C4—41no—
285C3×C4—41no—
286C3×C4—62no—
287C3×C4—62no—
288C3×C4—92no—
289C3×C4—122no—
290C3×C4D3×Dic612yes—
291C3×C4#379711yescharacteristic
292C3×C4#380211yescharacteristic
293C3×C4D3×#6712yes—
294Dic3—22no—
295Dic3—34no—
296Dic3—62no—
297Dic3—184no—
298Dic3—182no—
299Dic3—182no—
300Dic3—182no—
301Dic3—362no—
302Dic3—362no—
303Dic3—364no—
304C32—21no—
305C32—22no—
306C32—41no—
307C32#483912yescomplemented
308C32#511911yescharacteristic, complemented
309C23—91no—
310C2×C4—32no—
311C2×C4—91no—
312C2×C4—92no—
313C2×C4—92no—
314C2×C4—272no—
315C2×C4D3311yescharacteristic
316C8—62no—
317C8—182no—
318C2×C3—22no—
319C2×C3—22no—
320C2×C3—22no—
321C2×C3—21no—
322C2×C3—22no—
323C2×C3—42no—
324C2×C3—41no—
325C2×C3—182no—
326C2×C3#6380412yes—
327C2×C3#6380512yes—
328C2×C3#6381712yes—
329C2×C3D3×Dic3⋊C412yes—
330C2×C3#6390011yescharacteristic
331D3—64no—
332D3—122no—
333C22—92no—
334C22—182no—
335C22D3×#62411yescharacteristic
336C4—62no—
337C4—92no—
338C4—542no—
339C4#8895311yescharacteristic
340C4#8895711yescharacteristic
341C3—21no—
342C3—22no—
343C3—41no—
344C3#1059033211yescharacteristic, complemented
345C3#1059069512yescomplemented
346C2—182no—
347C2#1172282512yes—
348C2#1172291111yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutator, socleC22×C33C22×C4
Fitting, cosocleC2×C33×C4C23
prime radicalC22×C3×C3⋊D3C2×C4
hypocenterC33#424
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #11722050C2semidirect
5. C3×Dic3⋊(C3⋊C8)C2semidirect
30. Dic3⋊(C3⋊C8)D3semidirect
207. C33#424semidirect
307. C32#4839semidirect
308. C32#5119semidirect
344. C3#10590332semidirect
345. C3#10590695semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#888082
#117234372
C2×C3×#37122
#117220502
C3×Dic3⋊(C3⋊C8)2
#105906953
#105903323